We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert [28] and the canonical scattering diagrams of Gross-Hacking-Keel [24]. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.
We introduce a reduction from the distinct distances problem in R d to an incidence problem with (d − 1)-flats in R 2d−1 . Deriving the conjectured bound for this incidence problem (the bound predicted by the polynomial partitioning technique) would lead to a tight bound for the distinct distances problem in R d . The reduction provides a large amount of information about the (d − 1)-flats, and a framework for deriving more restrictions that these satisfy.Our reduction is based on introducing a Lie group that is a double cover of the special Euclidean group. This group can be seen as a variant of the Spin group, and a large part of our analysis involves studying its properties.
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